Conservation laws in classical mechanics for quasi-coordinates

Noether's theorem and Noether's inverse theorem for mechanical systems with gauge-variant Lagrangians under symmetric infinitesimal transformations and whose motion is described by quasi-coordinates are established. The existence of first integrals depends on the existence of solutions of the system of partial differential equations — the so-called Killing equations. Non-holonomic mechanical systems are analysed separately and their special properties are pointed out. By use of this theory, the transformation which corresponds to Ko Valevskaya first integral in rigid-body dynamics is found. Also, the nature of the energy integral in non-holonomic mechanics is shown and a few new first integrals for non-conservative problems are obtained. Finally, these integrals are used in constructing Lyapunov's function and in the stability analyses of nonautonomous systems. The theory is based on the concept of a mechanical system, but the results obtained can be applied to all problems in mathematical physics admitting a Lagrangian function.     

Hagedorn's theorem in some special cases of rheonomic systems

Hagedorn's theorem on instability [Arch. Rational Mech. Anal. 58 (1976) 1], deduced from Jacobi's form of Hamilton's principle, refers to scleronomic mechanical systems. In this paper we shall prove that Hagedorn's methodology can be generalized to a class of rheonomic mechanical systems with differential equations of motion which allow the existence of Painlevé's integral of energy. The application of this methodology to the case of rheonomic systems which allow, together with Painlevé's integral, cyclic integrals, as well as to the mechanical systems having resultant motion, with prescribed transport motion, and, finally, to the systems having Mayer's rheonomic potential, are also considered. Obtained results are illustrated by an example.     

One type of integrals for the equations of motion of higher-order nonholonomic systems

This paper presents one type of integrals and its condition of existence for theequations Of motion of higher-order nonholonomic systems,including I-order integral(generalized energy integral),2-order integral and p-order integral(P>2).All of theseintegrals can be constructed by the Lagrangianfunction of the system.Two examples aregiven to illustrate the application of the suggested method.     

A special case of integrability in a system with one quasi-cyclic coordinate

It is well known that conservative holonomic and scleromic systems with two degrees of freedom which have one cyclic coordinate are ‘integrable’. This means that the solution to the equations of motion can be given analytically in terms of quadratures, due to the existence of the two first integrals: the energy integral and the integral corresponding to the cyclic coordinate. In the present paper it is shown that the system is ‘integrable’ even if it is only holonomic and scleronomic and has one ‘quasi-cyclic’ coordinate, and even if the generalized forces are non-conservative provided the kinetic energy satisfies a certain additional condition.     

Transient stochastic response of quasi-partially integrable Hamiltonian systems

The approximate transient response of multi-degree-of-freedom (MDOF) quasi-partially integrable Hamiltonian systems under Gaussian white noise excitation is investigated. First, the averaged Itô equations for first integrals and the associated Fokker–Planck–Kolmogorov (FPK) equation governing the transient probability density of first integrals of the system are derived by applying the stochastic averaging method for quasi-partially integrable Hamiltonian systems. Then, the approximate solution of the transient probability density of first integrals of the system is obtained from solving the FPK equation by applying the Galerkin method. The approximate transient solution is expressed as a series in terms of properly selected base functions with time-dependent coefficients. The transient probability densities of displacements and velocities can be derived from that of first integrals. One example is given to illustrate the application of the proposed procedure. It is shown that the results for the example obtained by using the proposed procedure agree well with those from Monte Carlo simulation of the original system.     

Integrability of 3-dim polynomial systems with three invariant planes

We investigate the problem of integrability for a family of three-dimensional autonomous polynomial systems of ODEs. Necessary and sufficient conditions for the existence of two independent analytic first integrals for systems of the family are given. The linearizability of the systems is studied as well.     

Transient stochastic response of quasi integerable Hamiltonian systems

The approximate transient response of quasi integrable Hamiltonian systems under Gaussian white noise excitations is investigated. First, the averaged Ito equations for independent motion integrals and the associated Fokker-Planck-Kolmogorov (FPK) equation governing the transient probability density of independent motion integrals of the system are derived by applying the stochastic averaging method for quasi integrable Hamiltonian systems. Then, approximate solution of the transient probability density of independent motion integrals is obtained by applying the Galerkin method to solve the FPK equation. The approximate transient solution is expressed as a series in terms of properly selected base functions with time-dependent coefficients. The transient probability densities of displacements and velocities can be derived from that of independent motion integrals. Three examples are given to illustrate the application of the proposed procedure. It is shown that the results for the three examples obtained by using the proposed procedure agree well with those from Monte Carlo simulation of the original systems.     

On Approximations of First Integrals for Strongly Nonlinear Oscillators

In this paper strongly nonlinear oscillator equations will be studied.It will be shown that the recently developed perturbation method based onintegrating factors can be used to approximate first integrals. Not onlyapproximations of first integrals will be given, butit will also be shown how in a rather efficient way the existence and stability oftime-periodic solutions can be obtained from these approximations. In particularthe generalized Rayleigh oscillator equation will be studied in detail, and it will beshown that at least five limit cycles can occur.     

New stability criteria for stochastic perturbed singular systems in mean square

In this study, a generalized equal-peak principle is established to suppress the multimodal vibration of a multiple-degree-of-freedom (M-DOF) nonlinear system. Based on the proposed generalized principle, the design procedure of the multiple time-delayed vibration absorbers (TDVAs) is carried out. By four conditions in the proposed generalized principle, the objective of suppressing all the resonance peaks around multiple modes to the equal minimum values is realized. For the existence of nonlinearity, the necessary and sufficient conditions in the design procedure can guarantee that the two resonance peaks around each mode are simultaneously equal. Moreover, the two equal resonance peaks are suppressed to minimum values with the minimum peak condition. Two case studies verify the efficiency of the TDVAs designed by the generalized equal-peak principle for multimodal vibration suppression. Due to the multimodal vibration suppression capacity of the proposed TDVAs designed by the generalized equal-peak principle, significant broad frequency band vibration suppression effects are achieved. Thus, TDVAs and the proposed equal-peak principle have potential applications in the fields of high-DOF vibration systems, such as civil engineering, precision machining and aerospace.

     

基于哈密顿动力系统新变分原理的保辛算法之一：变分原理和算法构造

提出了哈密顿动力系统的一个新变分原理,并基于此变分原理构造了四类保辛算法。通过新的变分原理定义修正作用量,然后将位移和动量采用拉格朗日多项式近似,并采用高斯积分对时间近似积分得到近似的修正作用量。在修正作用量的基础上,通过选择时间步两端不同的位移或动量作为独立变量,可构造四种不同类型的保辛算法。     

Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems

An n degree-of-freedom Hamiltonian system with r(1<r<n) independent first integrals which are in involution is called partially integrable Hamiltonian system and a partially integrable Hamiltonian system subject to light dampings and weak stochastic excitations is called quasi partially integrable Hamiltonian system. In the present paper, the averaged Itô and Fokker-Planck-Kolmogorov (FPK) equations for quasi partially integrable Hamiltonian systems in both cases of non-resonance and resonance are derived. It is shown that the number of averaged Itô equations and the dimension of the averaged FPK equation of a quasi partially integrable Hamiltonian system is equal to the number of independent first integrals in involution plus the number of resonant relations of the associated Hamiltonian system. The technique to obtain the exact stationary solution of the averaged FPK equation is presented. The largest Lyapunov exponent of the averaged system is formulated, based on which the stochastic stability and bifurcation of original quasi partially integrable Hamiltonian systems can be determined. Examples are given to illustrate the applications of the proposed stochastic averaging method for quasi partially integrable Hamiltonian systems in response prediction and stability decision and the results are verified by using digital simulation.     

Stochastic Stabilization of Quasi-Partially Integrable Hamiltonian Systems by Using Lyapunov Exponent

A procedure for designing a feedback control to asymptoticallystabilize, with probability one, a quasi-partially integrableHamiltonian system is proposed. First, the averaged stochasticdifferential equations for controlled r first integrals are derived fromthe equations of motion of a given system by using the stochasticaveraging method for quasi-partially integrable Hamiltonian systems.Second, a dynamical programming equation for the ergodic control problemof the averaged system with undetermined cost function is establishedbased on the dynamical programming principle. The optimal control law isderived from minimizing the dynamical programming equation with respectto control. Third, the asymptotic stability with probability one of theoptimally controlled system is analyzed by evaluating the maximalLyapunov exponent of the completely averaged Itô equations for the rfirst integrals. Finally, the cost function and optimal control forces aredetermined by the requirements of stabilizing the system. An example isworked out in detail to illustrate the application of the proposedprocedure and the effect of optimal control on the stability of thesystem.     

On the General Structure of Small on Large Problems for Elastic Deformations of Varga Materials I: Plane Strain Deformations

In three recent papers [6–8], the present authors show that both plane strain and axially symmetric deformations of perfectly elastic incompressible Varga materials admit certain first integrals, which means that solutions for finite elastic deformations can be determined from a second order partial differential equation, rather than a fourth order one. For plane strain deformations there are three such integrals, while for axially symmetric deformations there are two. The purpose of the present papers is to present the general equations for small deformations which are superimposed upon a large deformation, which is assumed to satisfy one of the previously obtained first integrals. The governing partial differential equations for the small superimposed deformations are linear but highly nonhomogeneous, and we present here the precise structure of these equations in terms of a second-order linear differential operator D², which is first defined by examining solutions of the known integrals. The results obtained are illustrated with reference to a number of specific large deformations which are known solutions of the first integrals. For deformations of limited magnitude, the Varga strain-energy function has been established as a reasonable prototype for both natural rubber vulcanizates and styrene-butadiene vulcanizates. Plane strain deformations are examined in this present part while axially symmetric deformations are considered in Part II [16]. This revised version was published online in August 2006 with corrections to the Cover Date.     

FIRST INTEGRALS AND INTEGRAL INVARIANTS FOR VARIABLE MASS NONHOLONOMIC SYSTEM IN NONINERTIAL REFERENCE FRAMES

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Theory of generalized rays for SH-waves in dipping layers

The theory of generalized rays is applied to analyze transient waves in a layered half-space with non-parallel interfaces. The propagation, transmission, reflection, and refraction of SH waves which are generated by a line source in the surface layer of a three-layer model are considered, each of the two overlaying layers having a different dipping angle.Generalized ray integrals for multi-reflected rays in the top layer and for rays that are transmitted into the lower layer and then refracted back into the top layer are formulated by using three rotated coordinate systems, one for each interface, and are expressed in terms of local wave slowness along each interface. Through a series of transformations of the local slownesses, all ray integrals are expressible in a common slowness variable. The arrival time of each ray undergoing multiple reflections and transmissions is then determined from the stationary value of the phase function with common slowness of the ray integral. Inverse Laplace transform of these ray integrals are completed by Cagniard's method.     

Inverse Approach in Ordinary Differential Equations: Applications to Lagrangian and Hamiltonian Mechanics

This paper is on the so called inverse problem of ordinary differential equations, i.e. the problem of determining the differential system satisfying a set of given properties. More precisely we characterize under very general assumptions the ordinary differential equations in \(\mathbb {R}^N\) which have a given set of either \(M\) partial integrals, or \(M first integral, or \(M partial and first integrals. Moreover, for such systems we determine the necessary and sufficient conditions for the existence of \(N-1\) independent first integrals. We give two relevant applications of the solutions of these inverse problem to constrained Lagrangian and Hamiltonian systems respectively. Additionally we provide the general solution of the inverse problem in dynamics.     

Front Speed Enhancement by Incompressible Flows in Three or Higher Dimensions

We study, in dimensions N ≥ 3, the family of first integrals of an incompressible flow: these are ${H^{1}_{\rm loc}}$ functions whose level surfaces are tangential to the streamlines of the advective incompressible field. One main motivation for this study comes from earlier results proving that the existence of nontrivial first integrals of an incompressible flow q is the main key that leads to a “linear speed up” by a large advection of pulsating traveling fronts solving a reaction–advection–diffusion equation in a periodic heterogeneous framework. The family of first integrals is not well understood in dimensions N ≥ 3 due to the randomness of the trajectories of q and this is in contrast with the case N = 2. By looking at the domain of propagation as a union of different components produced by the advective field, we provide more information about first integrals and we give a class of incompressible flows which exhibit “ergodic components” of positive Lebesgue measure (and hence are not shear flows) and which, under certain sharp geometric conditions, speed up the KPP fronts linearly with respect to the large amplitude. In the proofs, we establish a link between incompressibility, ergodicity, first integrals and the dimension to give a sharp condition about the asymptotic behavior of the minimal KPP speed in terms of the configuration of ergodic components.     

Two degree of freedom gyroscopic systems with linear integrals

In this paper we consider the first integrals, linear in velocities, of conservative gyroscopic systems with two degrees of freedom. A precise criterion which specifies whether a given gyroscopic system possesses a linear integral is derived. When the kinetic energy has the structure of a standard form of the metric of revolution, all the possible systems which admit a linear integral and corresponding integrals are determined explicitly. Two examples are considered to illustrate the usefulness of the derived results.     

On stability of stationary motion of a nonconservative nonholonomic rheonomic system

One considers, in this paper, the motion of a mechanical system in a nonstationary field of potential and positional forces, subject to the action of rheonomic holonomic and nonholonomic linear homogeneous constraints. Assuming that differential equations of motion of the system considered satisfy the conditions for the existence of Painlevé's integral of energy, formulated in [Painlevé, P., 1897. Leçons sur l'intégration des équations de la Mécanique, Paris] and [Appell, P., 1911. Traité de mécanique rationnelle, T. II, Dynamique des systémes – Mécanique analitique, Gauthier-Villars, Paris] and generalized in [Čović, V., Vesković, M., 2004. On stability of motion of a rheonomic system in the field of potential and positional forces, BAMM-1720/2004, No-2233, 93–100] and [Čović, V., Vesković, M., 2005. Hagedorn's theorem in some special cases of rheonomic systems. Mechanics Research Communications 32 (3), 265–280], the original mechanical system is substituted by an equivalent one whose Lagrangian function, nontransformed with respect to nonholonomic constraints, does not depend on time explicitly. Using the properties of the equivalent system, which, in contrast to the original one, moves in a stationary field of potential forces and in a nonstationary field of gyroscopic forces, the definition of cyclic coordinates is generalized, as well as sufficient conditions for the existence of (cyclic) first integrals, corresponding to coordinates mentioned and linear in velocities are established. Further, the conditions for the existence of steady motion of the system considered are found. In the case of existence of such a motion of the system, the Theorem of Routh's type on stability of that motion, based on the minimum of reduced potential for which it is shown that, in contrast to known cases (see, for example, [Gantmacher, F., 1975. Lectures in Analytical Mechanics. Mir Publisher, Moscow; Neimark, J., Fufaev, N., 1972. Dynamics of Nonholonomic Systems. Amer. Math. Soc., Providence, RI; Pars, L., 1962. An Introduction to Calculus of Variations. Heinemann, London; Karapetyan, A., Rumyantsev, V., 1983. Stability of conservative and dissipative systems. In: Itogi Nauki I Tekhniki: Obschaya Mekh., vol. 6, VINITI, Moscow, pp. 3–128 (in Russian)]), it includes the influence of the positional forces field, is formulated. Thus, the Routh's Theorem on stability of steady motion of a conservative mechanical system is extended to the case of a nonconservative system.